Depletion-Mediated Potentials and Phase Behavior for Micelles

Sep 5, 2012 - Diffusing colloidal probes of cell surfaces. Gregg A. Duncan , D. Howard Fairbrother , Michael A. Bevan. Soft Matter 2016 12 (21), 4731-...
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Depletion-Mediated Potentials and Phase Behavior for Micelles, Macromolecules, Nanoparticles, and Hydrogel Particles Tara D. Edwards and Michael A. Bevan* Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States S Supporting Information *

ABSTRACT: We report a simple depletion potential that captures measured potentials and phase behavior for micrometer-sized colloids in the presence of unadsorbing charged micelles, charged nanoparticles, nonionic macromolecules, and nonionic hydrogel particles. Total internal reflection microscopy (TIRM) is used to measure net potentials between colloids and surfaces, and video microscopy (VM) is used to measure quasi-2D phase behavior in the same material systems. A modified Asakura−Oosawa (AO) depletion potential is developed to accurately quantify particle−wall potentials and interfacial crystallization via particle−particle potentials in Monte Carlo (MC) simulations. The modified AO potential includes effective depletant sizes, accurate osmotic equations of state, and partition coefficients. Partition coefficients are used as the sole adjustable fitting parameter, although an approach to their theoretical prediction from depletant density profiles is also presented. Our results demonstrate a model that accurately captures depletion interactions and phase behavior in a variety of material systems.

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INTRODUCTION The presence of nonadsorbing molecular assemblies, nanoparticles, and macromolecules in colloidal dispersions is known to induce depletion interactions between colloidal particles. Depletion interactions have been studied extensively in complex fluid mixtures of synthetic material components1,2 and are speculated to be widely prevalent in biological systems.3 As a result, understanding depletion attraction between colloidal components due to the presence of various nonadsorbing species provides a basis to manipulate colloidal dispersion microstructure, phase behavior, stability,4,5 and resulting material properties (e.g., rheology, optical properties). Despite many studies of depletion attraction and associated phenomena (too numerous to exhaustively review here), there does not yet exist a universal approach to model depletion attraction in the wide range of material systems in which such interactions are known to occur. Depletion attraction originates from the exclusion of nonadsorbing species from regions between particles and surfaces (i.e., excluded volume). The exclusion of nonadsorbing species (i.e., depletants) produces a local osmotic pressure (i.e., chemical potential difference) that results in an effective attraction between particles and surfaces (see Figure 1). The basic features of this attraction are captured by the Asakura− Oosawa (AO) potential,6 which is based on an ideal solution osmotic pressure and an excluded volume, computed by treating depletants as hard spheres. However, realistic depletants, such as charged micelles, charged nanoparticles, and macromolecules in good solvents, interact with each other to produce nonideal solution osmotic pressures. Likewise, such depletants interact with each other, colloids, and surfaces to © 2012 American Chemical Society

Figure 1. (A) (Left) Schematic of charged 2.34 μm SiO2 colloidal particles with particle−wall or particle−particle surface separation, h, experiencing depletion attraction with each other and a glass microscope slide due to the exclusion of spherical depletants (purple) with radius Lcore. Depletants can approach the SiO2 surfaces within a distance, ΔEV. Depletants can approach each other within a distance, ΔOP. (B) (Right, top-to-bottom) Schematic of different depletant types: (i) charged micelle (yellow), (ii) charged nanoparticle (gray), (iii) nonionic macromolecule (green), and (iv) nonionic hydrogel (cyan).

produce inhomogeneous concentration profiles between colloids and surfaces that are not generally captured by a simple hard sphere excluded volume. To overcome these limitations, a number of theoretical methods have been developed to model different depletant types. Despite the demonstrated accuracy of some of these approaches, they are Received: July 11, 2012 Revised: August 27, 2012 Published: September 5, 2012 13816

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Table S2 (Supporting Information), where the average partition coefficient, ⟨K⟩, was the only parameter adjusted in an iterative inverse analysis. All 1D simulations were performed for 3 × 104 MC steps. Noise was added to simulated particle elevations in a manner that accurately captures the TIRM measurement process. Specific aspects were described in detail in a recent paper.12 In short, Gaussian noise was added to MC simulation height excursions, which were then analyzed using the standard TIRM/VM analysis.10 Two-dimensional (2D) MC simulations of particle−particle interactions for comparison with VM measurements were performed for 256 log-normal size distributed colloids with a mean diameter reported in Table S2 (Supporting Information) and a relative standard deviation of 1.04 for 1% and 1.5% nanoparticle volume fractions, ϕ, and 1.01 for 2% and 3% nanoparticle volume fractions.13,14 All 2D simulations were performed for 10 × 106 MC steps using particle area fractions and initial configurations measured in VM experiments. Radial distribution functions, g(r), and 6-fold connectivity order parameters, ⟨C6⟩, were constructed from simulated particle coordinates with Gaussian noise added via a random number generator to produce uncertainty in particle center positions with a standard deviation of 4% the particle radius. The computation of ⟨C6⟩ is described in detail in several of our previous papers.7,9,15−17

often difficult to implement and are specific to different depletant materials. In this paper, we report measured depletion potentials and depletion-mediated quasi-2D phase behavior for four different depletant types, all of which we model using a simple depletion potential. In general, direct measurements of depletion potentials remain limited in number, are usually performed for single depletant types, are interpreted using material specific models, and are not typically connected to phase behavior measurements or predictions. To our knowledge, this work is the first to comprehensively consider depletion potentials and phase behavior in four different material systems. We use total internal reflection microscopy (TIRM) to directly measure depletion potentials and video microscopy (VM) to measure quasi-2D phase behavior for 2.34 μm SiO2 particles near glass microscope slides in the presence of nonadsorbing (see Figure 1) (A) charged sodium dodecyl sulfate (SDS) micelles,7 (B) charged SiO2 nanoparticles, (C) nonionic poly(ethylene oxide) (PEO) macromolecules,8 and (D) nonionic poly(N-isopropylacrylamide) (PNIPAM) hydrogel particles.9 In the following, we report a simple, modified AO potential that captures directly measured potentials and depletion-mediated phase behavior in each of these material systems. Although we have been developing such a potential as part of some of our previous work,7,8 we report a form here for the first time that accurately captures all of the depletion measurements we have made to date (i.e., both new and previous measurements).

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THEORY

Net Interaction Potential. The net potential energy for colloidal particles interacting with each other, an underlying surface, and gravity can be modeled as the sum of independent contributions. For charged colloids with radius a, in the presence of nonadsorbing depletant particles, the net interaction potential is given by,

MATERIALS AND METHODS

Materials. Nominal 2.34 μm diameter SiO2 colloids (Bangs Laboratories, Fishers, IN), Ludox grade AM-30 colloidal silica nanoparticles (Sigma-Aldrich Co., St. Louis, MO), and sodium chloride (NaCl, Acros Organics, Morris Plains, NJ) were used as received. Microscope slides (Fisher Scientific, Pittsburgh, PA) were sonicated in methanol (Fisher Scientific, Pittsburgh, PA) followed by acetone (Fisher Scientific, Pittsburgh, PA) for 30 min each, immersed in Nochromix (Godax Laboratories, Takoma Park, MD) for 1 h, soaked in 0.1 M KOH (Fisher Scientific, Pittsburgh, PA) for 30 min, washed with deionized (DI) water, and dried with nitrogen prior to each experiment. Dispersions of SiO2, Ludox, NaCl, and DI water, were prepared to yield 1.5 mM NaCl with ∼1% SiO2 interfacial area fractions for TIRM and ∼42% SiO2 interfacial area fractions for VM measurements. Microscopy. Experiments with Ludox nanoparticles were performed in sample cells consisting of 10 and 5 mm i.d. Vinton orings (McMaster Carr, Robbinsville, NJ) epoxy sealed between a microscope slide and a glass coverslip (Corning, Corning, NY) for TIRM and VM measurements, respectively. A 12-bit CCD camera (ORCA-ER, Hamamatsu, Japan) on an upright optical microscope (Axio Imager A1m, Zeiss, Germany) was used for TIRM and an inverted optical microscope (Axio Observer A1, Zeiss, Germany) was used for VM. For VM measurements, the camera was operated in onebinning mode with a 63× objective and 1.6× magnifying lens to yield 9 frames/s and 60 nm/pixel. For TIRM experiments, the camera was operated in four-binning mode with a 40× objective to yield 28 frames/s and 607 nm/pixel. TIRM experiments employed a 15mW 632.8 nm helium−neon laser (Melles Griot, Carlsbad, CA) and a 68° dovetail prism (Red Optronics, Mountain View, CA) to generate an evanescent wave decay length of β−1 = 114 nm. Image analysis algorithms coded in FORTRAN were used to track colloid lateral motion in TIRM and VM experiments and to integrate the evanescent wave scattering intensity from each colloid in TIRM experiments.10,11 Simulations. One-dimensional (1D) Monte Carlo (MC) simulations of particle−wall interactions were performed to include effects of noise for comparison with TIRM measurements. Simulations were performed using the theoretical potentials in eq 1 and parameters in

u(z , r ) = uGpf (z) + uEpp(r ) + uEpw (z) + uVpp(r ) + uVpw (z) + uSpp(r ) + uSpw (z) + uDpp(r ) + uDpw (z)

(1)

where z is the particle center-to-surface elevation relative to contact with the underlying planar surface at z = a, and r is the particle center-to-center separation relative to contact at r = 2a. The colloid−wall surface-to-surface separation, h = z − a (where z = h + a), or the colloid−colloid surface to surface separation, h = r − 2a (where r = h + 2a), can also be substituted into any particle−wall or particle−particle potential as a useful separation scale for potential interactions. Subscripts refer to interactions as (E) electrostatic, (V) van der Waals, (G) gravitational, (S) steric, and (D) depletion. Superscripts refer to interactions as (pp) particle−particle, (pw) particle−wall, and (pf) particle−field. Gravitational Potential. The gravitational potential energy of each particle depends on its elevation above the underlying surface multiplied by its buoyant weight, G, given by uGpf (z) = Gh = mg (z − a) = (4/3)πa3(ρp − ρf )g (z − a) (2)

where m is buoyant mass, g is acceleration due to gravity, and ρp and ρf are the particle and fluid densities. Derjaguin Approximation. The interaction potential between two spherical particles of equal radius, upp X (r), or between a particle and wall, upw X (z), can be obtained from the interaction energy per unit area between two flat plates (i.e., two walls), EXww(l), at separation l using the Derjaguin approximation as,18 13817

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uXpp(r ) = πa

∫r

uXpw (z) = 2πa

∞

∫z

Modified AO Depletion Potentials. The depletion attraction between particles and planar surfaces for all depletant material systems investigated can be given by a modified form of the usual AO depletion potential as

E Xww (l) dl

∞

E Xww (l) dl

(3)

uD(h , ΔEV ) =−V (h , ΔEV )ΔΠ

which is expected to be asymptotically exact as h/a → 0 and indicates that particle−wall interactions are the twice the value of particle−particle interactions at all separations (when the Derjaguin approximation is valid). Electrostatic Interaction Potentials. In all of the depletant material systems investigated, the colloidal particles are electrostatically stabilized against aggregation and deposition. The interaction between electrostatic double layers on two plates (from superposition, nonlinear Poisson−Boltzmann equation, 1:1 monovalent electrolyte)19 can be used in eq 3 to give analytical potentials for the interaction between particles and wall surfaces as20

= −V (h , ΔEV )ρo kT[Zo − Zi⟨K ⟩]

where V(h,ΔEV) is the usual excluded volume term for either particle−particle or particle−wall geometries,8,9,27 ΔEV is an effective depletant radius, ΔΠ is the osmotic pressure difference outside and inside the excluded volume region, ρo is the depletant number density outside the excluded volume region, Zo and Zi are compressibility factors evaluated at the average number densities outside and inside the excluded volume region, and ⟨K⟩ is an average partition coefficient defined as ⟨ρi⟩/ρo (where ⟨ρi⟩ is an average number density inside the excluded volume region). The usual excluded volume terms obtained from geometric considerations are given by19,28,29

uEpp(r ) = B exp[−κ(r − 2a)] uEpw (z) = 2B exp[−κ(z − a)]

V pp(r , ΔEV ) = π[(4/3)(a + ΔEV )3 (1 − (3/4)r(a + ΔEV )−1 + (1/16) r 3(a + ΔEV )−3 )]

⎛ kT ⎞2 ⎛ eψ ⎞ ⎛ eψ ⎞ B = 32πεa⎜ ⎟ tanh⎜ 1 ⎟ tanh⎜ 2 ⎟ ⎝ 4kT ⎠ ⎝ 4kT ⎠ ⎝ e ⎠ κ = (2e 2NAC /εkT )1/2

V pw(z , ΔEV ) = π[(4/3)ΔEV 3 + 4ΔEV 2 a − 4ΔEV a(z − a) + a(z − a)2 − ΔEV (z − a)2 + (1/3)(z − a)3 ]

(4)

where κ is the inverse Debye length, ε is the solvent dielectric constant, k is Boltzmann’s constant, T is absolute temperature, C is the 1:1 monovalent electrolyte molarity, NA is Avogadro’s number, e is the elemental charge, and ψ1 and ψ2 are the surface potentials on interacting surfaces. van der Waals Interaction Potentials. van der Waals attraction between flat plates are predicted from the rigorous Lifshitz theory21 (that includes retardation and screening effects) for flat plates as E Vww (l) = −A(l)/12πl 2

(10)

using the effective depletant size, ΔEV, reported in the text that accurately captures the range of the depletion attraction in all cases. To account for nonideal solution osmotic pressures, an accurate equation of state is used in the calculation of the osmotic pressure in eq 9 by incorporating an appropriate compressibility factor. The Carnahan−Starling30 (CS) equation of state is used for the charged micelle, charged nanoparticle, and nonionic hydrogel depletants and is given as

(5) 22

where A(l) is the Hamaker function. Equations 3 and 5 provide particle and wall van der Waals potentials, which can be represented over the separation and energy ranges of interest by convenient power law expressions as23,24

ZCS = (1 + ϕeff + ϕeff 2 − ϕeff 3)(1 − ϕeff )−3

(6)

where A and p are obtained from fits to the exact results. Adsorbed Polymer Brush Interaction Potentials. Steric repulsion due to the compression of excluded volume macromolecular brushes were modeled using the interaction per area between flat plates (from the well-established theory of Milner et al.)25,26 ESww (l)

=

Z RG

f (l) = f0 (5/9)[(δ0/l) + (l /δ0)2 − (1/5)(l /δ0)5 ]

⎛ 1 + 2.509ρ* + 1.360ρ*2 ⎞0.309 = 1 + 5.505ρo R G ⎜ ⎟ 1 + 0.596ρ* ⎠ ⎝ 3

(12)

where ρ* = 3.584ροRG3.

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RESULTS AND DISCUSSION Particle−Wall Potential Profiles. To demonstrate the universal behavior of a number of different depletant types, Figure 2 reports particle−wall potentials measured with ensemble TIRM.10,11 Particle−wall potentials are shown for all material systems described in the Introduction and Figure 1. The particles are electrostatically stabilized against aggregation and deposition in a 1:1 electrolyte solution. In the case of the PEO depletant, irreversible copolymer adsorption to silanized silica colloid surfaces produce PEO brushes, which prevent further PEO adsorption and generate steric repulsion.32 The gravitational potential energy has been subtracted from each

2(f (l /2) − f (δ0)), l ≤ 2δ0 0, l > 2δ0

(11)

where ϕeff is the effective depletant volume fraction. For excluded volume polymer depletants, the values of Π in eq 9 were computed using an equation of state from renormalization group31 (RG) theory based on the polymer radius of gyration, RG, and the polymer mass concentration,8 given as

uVpp(r ) = −aA[r − 2a]−p uVpw (z) = −a2A[z − a]−p

(9)

(7) (8)

where f(l) is the free energy per unit area of a brush compressed to a height, l < δ0, δ0 is the uncompressed brush layer thickness, and f 0 is the free energy (with units of kT) per unit area of an uncompressed brush. The expressions in eqs 7 and 8 were used in eq 3 to model interactions between macromolecular brushes adsorbed to particle and wall surfaces. 13818

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der Waals potentials are identical to the ones we have used in numerous papers using 2.34 μm SiO2 particles and glass microscope slides7−11,15,24,33 and produce excellent fits to the data at depletant concentrations too low to produce a detectable depletion attraction (i.e., CSDS < 8.5 mM, ϕSiO2 < 0.2%, CPEO < 0.214 g/L, ϕPNIPAM < 22%). Using eq 9, we are able to capture the directly measured potentials in the four depletant material systems using a simple, but modified, form of the original AO potential. We refer to eq 9 as a modified AO potential because in addition to the original theory it includes (1) an effective depletant radius, ΔEV, to account for soft depletants in V(h,ΔEV); (2) a compressibility factor, Z, to account for nonideal solution osmotic pressures; and (3) a partition coefficient, ⟨K⟩, to account for partial depletion from V(h,ΔEV). We have found values for each of the terms in eq 9 that quantitatively capture all of the depletion potentials for the different material systems in Figure 2. To accurately capture the range of the depletion attraction in all cases, the value of ΔEV, in V(h,ΔEV) was set equal to (1) for the PNIPAM depletants, the hydrodynamic radius, Lhydro, from dynamic light scattering;9 (2) for PEO macromolecules, an order unity factor34,35 multiplied by the density correlation length, ξ, from static light scattering measurements of the radius of gyration, RG,25 and theoretical predictions31 (plus an estimated distance of closest approach to nonadsorbing surfaces, λ, due to interactions with adsorbed polymer brushes); (3) for the charged micelles, their core size, Lcore, from neutron scattering36 and fluorescence37 measurements, plus 4.7κ−1 to account for electrostatic interactions;38,39 and (4) for the silica nanoparticles, their core size, Lcore, from dynamic light scattering, plus 4.7κ−1 to account for electrostatic interactions.38,39 The importance of the electrostatic repulsion between the charged depletants and the charged SiO2 colloids and wall surfaces is reflected in the effective depletant radius in that it depends on the hard sphere depletion radius, Lcore, and the addition of some factor multiplied by κ −1. This factor can be estimated from an energetic cutoff in the depletant−particle (and/or depletant−wall) interaction potential.20 We used a value of 4.7κ−1 as was previously reported in similar experimental19 and modeling studies20 concerned with charged depletants. This value was found to accurately capture the range of our measured depletion potential for both the micelle and nanoparticle depletants. To quantitatively capture the depth of the depletion attraction in all cases in Figure 2, it is necessary to have an accurate equation of state and to account for equilibrium partitioning of depletants between bulk and excluded volume regions. The osmotic equation of state was chosen for each depletant on the basis of their material properties. For the PEO depletants, the values of Π in eq 9 were computed using an

Figure 2. Particle−wall potentials from TIRM measurements of (A) SDS micellar depletants for 1 mM (blue), 6 mM (red), 8.5 mM (green), 10 mM (pink), 12 mM (cyan), 14 mM (yellow), and 16 mM (gray) SDS concentrations from Iracki et al.27 (B) SiO2 nanoparticle depletants with 0.2% (blue), 0.4% (red), 0.6% (green), 0.8% (pink) volume fractions. (C) PEO macromolecular depletants for 5 mM NaCl (blue) and 0.214 g/L (red), 0.321 g/L (green), 0.428 g/L (pink), 0.642 g/L (cyan), and 0.856 g/L (yellow) PEO concentrations from Edwards and Bevan.8 (D) hydrogel depletants for 22% (20 °C, red), 24% (22 °C, blue), 26% (24 °C, green), 28% (26 °C, pink), 31% (27 °C, cyan), 34% (28 °C, yellow), 36% (28.5 °C, gray), and 37% (29 °C, purple) PNIPAM particle volume fractions from Fernandes et al.9 Insets depict each depletant type according to the color themes in Figure 1.

potential, since it is a simple linearly varying body force (rather than a surface force contributing to the particle−wall interaction). The depth of the attractive well in each case is mediated by depletion attraction, which is determined by the depletant volume fraction. The depletant volume fraction in Figure 2A−C is determined by the depletant concentration, whereas the depletant volume fraction in Figure 2D is determined by the PNIPAM temperature-dependent size at a fixed concentration (see Table 1). In each case, the depletion attraction produces a single well with a monotonically increasing well depth and decreasing minimum location with increasing depletant volume fraction. The data are accurately captured by theoretical potentials in Figure 2 given by the superposition of electrostatic, steric, van der Waals, and depletion potentials, and parameters reported in the Supporting Information. The electrostatic, steric, and van

Table 1. Depletant Properties, Physical System Values, and Equations of State (EOS) Used for Theoretical Fits To Calculate Depletion Interaction Potentials (eq 9) in TIRM (Figure 2) and in MC Simulations (Figure 3) for Each Depletant Material System

Lcore/nm ΔEV/nm ΔOP/nm f equation of state

micelle

nanoparticle

macromolecule

hydrogel

1.8−2.4 Lcore + 4.7κ−1 = 16.0−18.1 Lcore + fκ−1 = 3.8−4.4 0.63−0.76 CS

6.0 Lcore + 4.7κ−1 = 25.3−39.8 Lcore + fκ−1 = 11.3−15.2 1.283−1.292 CS

ξ = 23.2−26.1 1.9ξ/√π + λ = 30.1−34.2 − − RG

− Lhydro = 113.5−81.5 Lhydro = 113.5−81.5 − CS

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equation of state from RG31 theory based on RG and the polymer mass concentration.8 In all other cases, the values of Π in eq 9 were computed using the Carnahan−Starling30 (CS) equation of state based on the volume fraction, ϕ = (4/ 3)πΔOP3ρo, for PNIPAM and the effective volume fraction, ϕeff = (4/3)πΔOP3ρo, for charged micelles and nanoparticles. The PNIPAM ΔOP value was computed on the basis of the same value in the excluded volume term, ΔOP = ΔEV = Lhydro. In contrast to the charged effective depletant size used in the excluded volume term, ΔEV, which reflects the importance of electrostatic interactions between the charged depletants and colloids and wall surfaces, the charged effective hard sphere depletant size, ΔOP, used in the osmotic pressure term reflects the importance of electrostatic repulsion between the depletants with each other. As such, perturbation theory was used to approximate the interactions between charged depletants and their respective hard sphere effective sizes. Specifically, the charged depletants’ ϕeff values were computed using ΔOP=Lcore + fκ−1, where f was determined from perturbation theory.40 The values of f on average fall between 0.63 and 0.76 for the micelles and 1.283 and 1.292 for the nanoparticles. The partial exclusion of soft depletants was accounted for by ⟨K⟩ in eq 9, which we implemented as a concentrationdependent, but separation-independent, parameter. It should be noted that ⟨ρi⟩, and hence ⟨K⟩, are in general a function of separation and bulk concentration based on theoretical expectations16,39,41 (i.e., spatially inhomogeneous depletant concentration profiles at each particle−surface separation). Here we explore a minimally complex model, where ⟨K⟩ is an averaged (spatially and separation) constant assumed to depend only on ρo, the depletant number density outside the excluded volume region. When using ⟨K⟩ as the sole adjustable parameter in eq 9, excellent fits are obtained for all of the potentials in Figure 2. This confirms the ability of eq 9 to capture both the shape and magnitude of the interaction for all of the depletant types investigated here. Before discussing the fit ⟨K⟩ values, we first investigate the ability of eq 9 to capture depletion-mediated quasi-2D phase behavior. Depletion-Mediated Quasi-2D Phase Behavior. To further probe the validity of the potential in eq 9 for capturing depletion-mediated phase behavior in concentrated systems (beyond the direct measurements of pair potentials at near infinite dilution), Figure 3 reports results from both VM measurements and MC simulations of the same material systems measured in Figure 2. For each depletant type, Figure 3 shows (1) VM measured 2D fluid configurations, (2) VM measured 2D crystalline configurations, and (3) radial distribution functions, g(r), from both VM measurements and MC simulations for the conditions corresponding to the images. The images are processed to show whether bonds between adjacent particles are crystalline (blue) or amorphous (red). In each case, dynamic equilibrium fluid configurations are observed at lower depletant volume fractions, and crystalline configurations coexisting with mobile gas particles are observed at higher depletant volume fractions. At depletant volume fractions intermediate to the conditions in Figure 3, fluid/solid coexistence was observed, and at higher depletant volume fractions, amorphous and arrested gel configurations were observed. To determine the effective pairwise potentials between the micrometer-sized SiO2 colloids in Figure 3, MC simulations were run in an iterative fashion to match simulated and

Figure 3. (left-to-right) VM measured 2D colloidal configurations as a function of depletant volume fraction for (column 1) dynamic equilibrium quasi-2D fluids; (column 2) crystallites coexisting in dynamic equilibrium with fluid particles; and (column 3) radial distribution functions, g(r), from VM (points) and MC simulations (lines). The images in columns 1 and 2 are processed to show whether bonds between adjacent particles are crystalline (blue) or amorphous (red). Depletant volume fractions in g(r) correspond to those in column 1 (red, bottom) and those in column 2 (blue, top). Insets depict each depletant type according to the color themes in Figure 1.

measured g(r) in a sort of “inverse” analysis.13,14 In detail, the particle−particle depletion potentials are obtained using the following procedure (described in numbered sections). (1) First, MC simulations were performed using the same potentials used to fit the TIRM measurements except that appropriate geometric corrections for particle−particle interactions were included. As in the TIRM measurements, the electrostatic, steric, and van der Waals potentials were identical to those in our previous papers using 2.34 μm SiO 2 particles7−11,15,24,33 and are exactly half the particle−wall potentials via the Derjaguin approximation. In the depletion potential in eq 9, the only geometric modification is a different expression for V(h,ΔEV). All potentials also used the same parameters as the TIRM potential fits, except for ⟨K⟩, which remained as the only adjustable parameter. (2) Next, values of ⟨K⟩ were adjusted in the MC simulations. These simulations were also run at higher bulk depletant concentrations compared to Figure 2 because the different geometry produces a smaller excluded volume that requires higher bulk depletant concentrations to produce similar well depths (i.e., the sphere−sphere interaction is approximately half the sphere−wall interaction). (3) Then, the radial distribution functions (Figure 3), C6 13820

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solution osmotic pressures can lead to very different levels of depletion attraction for different combinations of depletant type and bulk volume fractions. Without a potential like the one in eq 9, these systems appear to display very different behavior that has previously been difficult to capture in a single model. Beyond the potential in eq 9, the trends in the ⟨K⟩ values in Figure 4 can be understood qualitatively in terms of a combination of depletant dimensions relative to distances of closest approach between particles and surfaces and the depletant “softness.” For example, the charged micellar and nanoparticle depletants can still physically fit within the gap between charged particles and surfaces that remain separated by electrostatic repulsion. Hence, notable depletant partitioning is detected in the charged depletant systems. This behavior arises from a continuously increasing energy penalty for inserting a charged depletant in the gap between charged surfaces with decreasing separation. Also, for a given surface separation, the soft interaction between depletants and surfaces also allows more depletants to enter the excluded volume region via greater osmotic pressures at higher bulk depletant volume fractions. The behavior of the charged depletants contrasts with the insertion of a hard sphere depletant between surfaces that also have hard wall interactions with each other and the depletant. In this case, there is discontinuous change from no energy difference to an infinite energy penalty at exactly the separation when the depletant no longer fits in the gap. Although the nonionic PNIPAM hydrogel particle is not exactly a hard sphere, it essentially either fits or does not fit in the gap between the silica colloids and surfaces. Therefore, the hydrogel particles essentially behave as effective hard sphere depletants, as reflected in their partition coefficients of ⟨K⟩ = 0 at all depletant volume fractions. The PEO macromolecular depletant is perhaps the most complicated case. As stated earlier, copolymer adsorption produces PEO brushes, which prevent further adsorption and allow unadsorbed PEO chains to act as depletants. When brushes on opposing surfaces come into contact, they can be expected to completely exclude unadsorbed PEO chains. However, at separations comparable to the effective PEO size, soft interactions of unadsorbed PEO random coils with each other and PEO surface brushes are expected to produce finite partitioning between the gap and bulk.16,42 Despite this theoretical expectation, the fit ⟨K⟩ values approach zero in all PEO experiments and suggest more complete exclusion of the PEO chains. Because fit ⟨K⟩ values are an average over a series of unique inhomogeneous PEO density profiles at each particle−particle or particle−wall separation, a significant amount of detail is lost about partitioning as a function of separation. In addition, since ⟨K⟩ is the sole adjustable parameter in eq 9, it compensates for the net uncertainty in other input quantities (e.g., equation of state, effective PEO dimensions), which could also account for ⟨K⟩ being lower than expected. Despite this minor concern, and in the absence of more direct measurements or rigorous theories of the PEO partitioning, the PEO partitioning is still qualitatively reasonable within expectations. Theoretical predictions of ⟨K⟩ are beyond the present paper’s immediate goal of simply showing that eq 9 is capable of capturing depletion potentials and phase behavior in a variety of material systems. However, we briefly suggest how ⟨K⟩ values could be obtained from more rigorous theoretical

histograms (not shown), and particle configurations (Figure 3) obtained from the MC results were compared with the VM measurements. (4) Finally, the ⟨K⟩ values were adjusted in the depletion potentials until the MC simulations matched the VM experiments. At the conclusion of this process, the converged ⟨K⟩ values and all effective pair potentials were known. As in the TIRM results in Figure 2, the primary conclusion is that the fit ⟨K⟩ values do an excellent job of capturing the phase behavior in Figure 3. The results in Figure 3 confirm the ability of eq 9 to capture quasi-2D phase behavior for all of the depletant types and concentrations investigated in this work. Depletant Partitioning in Excluded Volume Regions. Given that the modified AO potential in eq 9 captures all of the potentials and phase behavior in Figures 2 and 3 for a variety of materials systems, we now turn to a more thorough discussion of the adjustable parameter, ⟨K⟩, which enables this agreement. Figure 4 summarizes the ⟨K⟩ values for all materials systems

Figure 4. Average partition coefficient, ⟨K⟩, as a function of bulk effective depletant volume fraction, ϕeff, plotted on a log scale. Closed data points indicate values obtained from fits to direct particle−wall measurements in Figure 2, and open data points designate values obtained from indirect particle−particle measurements in Figure 3.

(micelles, nanoparticles, macromolecules, hydrogels) and experiments (potential and phase behavior measurements) in this paper. Specifically, Figure 4 reports fit values of ⟨K⟩ for each depletant type vs effective depletant volume fractions from both the TIRM measured potentials in Figure 2 and the VM measured phase behavior in Figure 3. Because the V(h,ΔEV) term in eq 9 already accounts for geometric differences in the particle−wall and particle−particle potentials, we propose that the ⟨K⟩ values from TIRM and VM correspond to the same quantity and can be compared on the same plot. The results show that ⟨K⟩ approaches zero for nonionic macromolecular and hydrogel depletants, which indicates nearly complete exclusion of depletants from the gaps between particles and wall surfaces. In other words, the nonionic depletant results approach the AO model of complete exclusion, although it is still necessary to assign effective depletant dimensions and to use accurate equations of state. In contrast, the charged micellar and nanoparticle depletants display very different behavior from the nonionic depletants. Specifically, significant partitioning is observed at low depletant bulk volume fractions, and very little partitioning, or exclusion, appears to occur at elevated bulk depletant volume fractions. Overall, Figure 4 shows that different degrees of exclusion and 13821

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Langmuir approaches. In particular, any theory capable of predicting depletant density profiles could be used to predict ⟨K⟩ values as

∫0

2ΔEV

h−1 dh

∫0

h

ρi (ρo , h , x) dx

(13)

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CONCLUSIONS Our direct measurements of potentials and phase behavior mediated by charged micelles, charged nanoparticles, nonionic random coils, and nonionic hydrogel depletants are welldescribed by a modified AO potential. The modifications to the original AO theory include an effective depletant size, an accurate osmotic equation of state, and an average partition coefficient to account for partial depletant exclusion. Our results show that effective depletant sizes and equations of state for each depletant type can be used without any adjustable parameters based on available theoretical models. The only adjustable parameter is the partition coefficient, ⟨K⟩, which produces excellent fits to all directly measured potentials and those inferred from phase behavior measurements for all four depletant types. Future work could obtain ⟨K⟩ from theories of depletant density profiles between surfaces. Although the current potential has not been shown to work for all depletant/ colloid size ratios and depletant concentrations, a separationdependent K(h) could also be specified to account for oscillatory potentials typical of other conditions not investigated in this work. For the case of single depletion wells on the order of kT, which are important to colloidal phase behavior and self-assembly, the modified AO potential accurately captures the range and magnitude of the directly measured potentials. Our results provide a foundation for establishing a simplified, universal model of depletion interactions and phase behavior in a variety of depletant material systems. ASSOCIATED CONTENT

* Supporting Information S

New data on SiO2 nanoparticle depletants, some additional theoretical expressions, and potential parameters for all depletant types and experiments. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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where ρi(ρo,h,x) is the density profile between particles (or particles and surfaces) and x is the position between the depletant particles and surfaces (at a given separation, h). Examples of predicted ρi(ρo,h,x) are available for hard spheres,41 charged particles,39 and macromolecular16 depletants, which are suitable for the depletants investigated in this work. Future work could include parametrized ⟨K⟩ functions via eq 13 that could be used in conjunction with eq 9 to model and predict depletion potentials for a broad range of material systems. Equation 13 also serves the purpose of showing the averaging that is inherently included as part of fitting constant ⟨K⟩ values to the measured potentials and potentials inferred from phase behavior measurements.

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ACKNOWLEDGMENTS

We acknowledge financial support by the National Science Foundation (CBET0932973).

K ≡ ρi (ρo , h , x)/ρo ⟨K ⟩ ≡ (2ΔEV ρo )−1

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 13822

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